Joint Source and Relay Matrices Optimization for Interference MIMO Relay Systems

Transcription

1 Joint Source and Relay Matrices Optimization for Interference MIMO Relay Systems Khoa Xuan Nguyen Dept Electrical & Computer Engineering Curtin University Bentley, WA 6102, Australia Yue Rong Dept Electrical & Computer Engineering Curtin University Bentley, WA 6102, Australia Abstract In this paper, we investigate the transceiver design for an amplify-and-forward interference multiple-input multipleoutput MIMO relay communication system The minimum mean-squared error MMSE of the signal waveform estimation is chosen as the design criterion to optimize the source, relay, and receiver matrices for interference suppression An iterative algorithm is proposed to solve the nonconvex source, relay, and receiver optimization problem Simulation results demonstrate that the proposed algorithm outperforms the existing technique in terms of both MSE and bit-error-rate Index Terms Interference channel, MIMO relay, MSE I INTRODUCTION Relay aided multiple-input multiple-output MIMO communication technology has attracted great research interest recently [1]-[2] By incorporating relay nodes in a MIMO system, the network coverage and reliability can be significantly improved In a MIMO relay system, communication between source nodes and destination nodes can be assisted by single or multiple relays equipped with multiple antennas The relays can either decode-and-forward DF or amplifyand-forward AF the relayed signals [3] In the AF scheme, the received signals are simply amplified including a possible linear transformation through the relay precoding matrices before being forwarded to the destination nodes Therefore, in general the AF strategy has lower complexity and shorter processing delay than the DF strategy For single-user two-hop MIMO communication systems with a single relay node, the optimal source and relay precoding matrices have been developed in [4] For a single-user twohop MIMO relay system with multiple parallel relay nodes, the design of relay precoding matrices has been studied in [5] Recent progress on the optimization of AF MIMO relay systems has been summarized in the tutorial of [2] For MIMO interference channel, the idea of interference alignment IA [6] was developed for interference suppression by arranging desired signal and interference into appropriated signal space The idea of IA has been applied in interference MIMO relay system in [7]-[8] However, there is still no general solution for IA as a number of conditions must be met One main reason is that the number of dimensions required for IA is very large and it depends on the number of independent fading channels This leads to high computational complexity and infeasibility in practical systems In [9], an iterative algorithm has been proposed to optimize the source beamforming vector and the relay precoding matrices to maximize the signal-to-interference-noise SINR at the destination nodes In this paper, we investigate the transceiver design for an AF interference MIMO relay communication system where multiple source nodes transmit information simultaneously to the destination nodes with the aid of multiple relay nodes, and each node is equipped with multiple antennas We aim at optimizing the source, relay, and receiver matrices to suppress the interference and minimize the mean-squared error MSE of the signal waveform estimation at the destination nodes, subjecting to transmission power conditions at source and relay nodes Since the original optimization problem is nonconvex with matrix variables, we propose an iterative algorithm In each iteration of the proposed algorithm, we first optimize all relay matrices based on the source and receiver matrices from the previous iteration Then we optimize all source matrices using the relay matrices in this iteration and the receiver matrices from the previous iteration Finally, the receiver matrices are updated Simulation results demonstrate that the proposed algorithm outperforms the existing technique in terms of both MSE and bit-error-rate Throughout this paper, scalars are denoted with lower or upper case normal letters, vectors are denoted with boldfaced lower case letters, and matrices are denoted with boldfaced upper case letters Superscripts T, H, and 1 denote transpose, conjugate transpose and inverse, respectively, tr stands for the trace of a matrix, vec stacks columns of a matrix on top of each other into a single vector, bd denotes a block-diagonal matrix, represents the Kronecker product, E[ ] denotes the statistical expectation, and I n stands for the n n identity matrix II SYSTEM MODEL AND PROBLEM FORMULATION We consider a two-hop interference MIMO relay communication system where K source-destination pairs communicate simultaneously with the aid of a network of L distributed relay nodes as shown in Fig1 Similar to [9], we ignore the direct links between source and destination nodes as they undergo much larger path attenuation compared with the links via relays The kth source node and the kth destination node are Copyright C 2014 by IEICE 640

2 B 1 B K H 1K H 11 H L1 H LK F 1 F L G 11 G 1L G KL G K1 W 1 W K Fig 1: Block diagram of an interference MIMO relay system equipped with N sk and N dk antennas, respectively, and the number of antennas at the lth relay node is N rl Using half duplex relay nodes, the communication between source and destination pairs is completed in two time slots At the first time slot, each source node transmits an N sk 1 signal vector x sk = B k s k, k = 1,,K 1 to the relay nodes, where s k is the d 1 information-carrying symbol vector and B k is the N sk d source precoding matrix The received signal vector at the lth relay node is given by y rl = H lk x sk v rl, l = 1,,L 2 whereh lk is then rl N sk MIMO channel matrix between the kth source node and the lth relay node and v rl is the N rl 1 additive white Gaussian noise AWGN vector at the lth relay node with zero mean and covariance matrix E[v rl v H rl ] = σ 2 rl I N rl, l = 1,,L The received signal vector at the lth relay node is amplified with the N rl N rl precoding matrix F l as x rl = F l y rl, l = 1,,L 3 The precoded signal vector x rl is forwarded to the destination nodes The received signal vector at the kth destination node is given by y dk = G kl F l y rl v dk, k = 1,,K 4 where G kl is the N dk N rl MIMO channel matrix between the lth relay node and the kth destination node and v dk is the N dk 1 AWGN vector at the kth destination node with zero mean and covariance matrix E[v dk v H dk ] = σ2 dk I N dk, k = 1,,K Due to their simplicity, linear receivers are used at the destination nodes to retrieve the transmitted signal Thus, the estimated signal vector at the kth destination node can be written as ŝ k = W H k y dk, k = 1,,K 5 wherew k is then dk d receiver weight matrix From 1-5, we have ŝ k =W H k =W H k L K G kl F l H lm B m s m v dk m=1 G kl F l H lk B k s k } {{ } desired signal G kl F l K H lm B m s m Wk H v dk 6 W H k }{{} interference plus noise 7 where v dk L G klf l v rl v dk is the total noise at the kth receiver From 1 and 3, the transmission power constraints at the source and relay nodes can be written as tr B k B H k Psk, k = 1,,K 8 tr F l E [ y rl yrl] H F H l Prl, l = 1,,L 9 where P sk and P rl denote the power budget at the kth source node and the lth relay node, respectively, and E [ y rl yrl] H = K m=1 H lmb m B H m HH lm σ2 rl I N rl is the covariance matrix of the received signal vector at the lth relay node In this paper, we aim at optimizing the source precoding matrices {B k } {B k,k = 1,,K}, the relay precoding matrices {F l } {F l, l = 1,,L}, and the receiver weight matrices {W k } {W k, k = 1,,K}, to minimize the sum-mse of the signal waveform estimation at the destination nodes under transmission power constraints at the source and relay nodes We would like to mention that minimal MSE MMSE is a sensible design criterion based on the links of MSE to other performance measures in MIMO systems such as mutual information and SINR [4], [10] From 7, the MSE of the kth source-destination pair can be calculated as MSE k =tr E [ŝ k s k ŝ k s k H] =tr Wk H H k I d Wk H H k I d H W H k C k W k W H k Ξ k W k, k = 1,,K 10 where H k is the equivalent MIMO channel matrix of the kth source-destination pair, C k = E [ v dk v H dk] and Ξk are the covariance matrices of the equivalent noise and the interference at the kth pair, respectively They are given respectively as H k = G kl F l Hlk, k = 1,,K L L H C k =E G kl F l v rl v dk G kl F l v rl v dk = σrlg 2 kl F l F H l G H kl σdki 2 Ndk, k = 1,,K Copyright C 2014 by IEICE 641

3 Ξ k = G kl F l Ξ k,l,n F H n G H kn, k = 1,,K n=1 where H lk H lk B k is the equivalent MIMO channel matrix between the kth source node and the lth relay node and Ξ k,l,n K H lm HH nm, k = 1,,K, l,n = 1,,L From 8-10, the optimal source, relay, and receiver matrices design problem can be written as min {W k },{B k },{F l } MSE k 11 st tr B k B H k Psk,,,K 12 tr F l E[y rl yrl]f H H l Prl,,,L13 III PROPOSED SOURCE, RELAY, AND RECEIVER MATRICES DESIGN ALGORITHM The problem is highly nonconvex with matrix variables, and a globally optimal solution is intractable to obtain In this section, we propose an iterative algorithm to solve the problem by optimizing {W k }, {B k }, and {F l } in an alternating way In each iteration of this algorithm, we first optimize {W k } based on {B k } and {F l } from the previous iteration Then we optimize all relay matrices based on {W k } from the current iteration and {B k } from the previous iteration Finally, we optimize all source matrices using {W k } and {F l } from the current iteration It can be seen from 10 the W k only affects MSE k Thus, with given {F l } and {B k }, the optimal linear receiver matrix which minimizes MSE k in 10 is the well-known MMSE receiver [11] given by W k = H k HH k C k Ξ k 1 Hk, k = 1,,K 14 With given receiver matrices {W k } and source precoding matrices {B k }, the sum-mse = K MSE k can be rewritten as a function of {F l } as L L H = tr Ḡ kl F l Hlk I d Ḡ kl F l Hlk I d σrlḡklf 2 l F H l Ḡ H kl σ2 dk WH k W k n=1 Ḡ kl F l Ξ k,l,n F H n ḠH kn 15 where Ḡkl Wk HG kl is the equivalent MIMO channel matrix between the lth relay node and the kth destination node Using the identities of [12] tra T B=vecA T vecb 16 tra H BAC=vecA H C T BvecA 17 vecabc=c T AvecB 18 the 15 can be represented as a function of f l vecf l, l = 1,,L, as [ L H L = O kl f l veci d O kl f l veci d n=1 fl H Q kl f l fn H ˆΞ T k,l,n ĜH knĝklf l ]t 1 19 where t 1 K σ2 dk trwh k W k is independent of f l, l = 1,,L, and for k = 1,,K, l = 1,,L O kl = H T lk Ḡkl, Q kl = σ 2 rli Nrl ḠH klḡkl 20 For k = 1,,K, let us introduce O k =[O k1, O k2,, O kl ] Q k =bdq k1, Q k2,, Q kl U k,ln =Ξ T k,l,n ḠH knḡkl U k,11 U k,1l U k = U k,l1 U k,ll Then the function 19 can be written as a function of f = [f T 1, ft 2,,fT L ]T as ψ 1 f= [ O k f veci d H O k f veci d By introducing f H Q k f f H U k f ] t 1 21 K T D l = H lm B m B H m HH lm σ2 rl I N r I Nrl, l = 1,,L m=1 and D l = bdd l1, D l2,,d ll, where D ll = D l and D lj = 0, l j, the relay transmit power constraint in 9 can be rewritten as f H Dl f P rl, l = 1,,L 22 From 21 and 22, the relay matrices optimization problem can be written as min ψ 1 f f 23 st f H Dl f P rl, l = 1,,L 24 The problem is a quadratically constrained quadratic programming QCQP problem [13], which is a convex optimization problem and can be efficiently solved by the interior-point method [13] The problem can be solved by the CVX MATLAB toolbox for disciplined convex programming [14] Copyright C 2014 by IEICE 642

4 With given receiver matrices {W k } and relay matrices {F l }, the sum-mse can be rewritten as a function of {B k } as L L H = tr Ḡ kl F l H lk B k I d Ḡ kl F l H lk B k I d n=1 Ḡ kl F l K H lm B m B H m HH lm FH n ḠH kl t 2 25 where t 2 K trwh k C kw k can be ignored in the optimization process as it is independent of {B k } Using the identities in 16-18, the function in 25 can be written as [ = S k b k veci d H S k b k veci d = b H m I d n=1 H H nmf H n ḠH knḡklf l H lm b m ]t 2 ] [S k b k veci d H S k b k veci d b Hk T k b k t 2 26 where for k = 1,,K S k I d T k I d Ḡ kl F l H lk n=1 H H nk FH n ḠH mnḡmlf l H lk By introducing T bdt 1,T 2,,T K and S k [S k1,s k2,,s kk ], where S kk = S k and S ki = 0, i k, the function 26 can be written as a function of b = [b T 1, bt 2,,bT K ]T as Φ 1 b = H Sk b veci d Sk b veci d b H Tbt 2 27 Let us introduce E ij = I d H H ij FH i F ih ij, El = bde l1,e l2,,e lk, Ēi = bd Ē i1,ēi2,,ēik, where Ē ii = I dns andēij = 0,i j The optimalbcan be obtained by solving the following problem min Φ 1 b b 28 st b H Ē m b P sm, m = 1,,K 29 b H E l b P rl σ 2 rltrf l F H l, l = 1,,L 30 The problem is a QCQP problem and can be solved by the CVX MATLAB toolbox [14] for disciplined convex programming The steps of applying the proposed iterative algorithm to optimize {B k }, {F l }, and {W k } are summarized in Table I, where the superscript n denotes the variable at the nth TABLE I: Procedure of solving the problem by the Proposed Algorithm 1 1 Initialize the algorithm with { F 0 l 9; Set n = 0 2 Obtain { W n1 k 3 Update {F n1 } { 0} and B k satisfying 8 and } } { n} { n based on 14 with fixed F l and B k } through solving the problem with given { l n} { n1} B k and W k 4 Update {B n1 k } by solving the problem with fixed { n1} { n1} F l and W k 5 If n n1 ε, then end Otherwise, let n := n1 and go to Step 2 iteration, and ε is a small positive number up to which convergence is acceptable Since all subproblems 11, 23-24, and are convex, the solution to each subproblem is optimal Thus, the value of the objective function 11 decreases after each iteration Moreover, the objective function is lower bounded by at least zero Therefore, the iterative algorithm converges to at least a locally optimal solution IV NUMERICAL EXAMPLES In this section, we illustrate the performance of the proposed algorithm through numerical simulations All channel matrices have independent and identically distributed iid complex Gaussian entries with zero-mean and unit variance The noises are iid Gaussian with zero mean and unit variance The QPSK constellations are used to modulate the source symbols For the sake of simplicity, we assume that all nodes have three antennas, ie, N sk = N dk = N rl = 3, k = 1,,K, l = 1,,L, all source nodes have the same power budget as P sk = P s = 15dB, k = 1,,K, and all relay nodes have the same power budget as P rl = P, l = 1,,L For all simulation examples, there are K = 3 source-destination pairs, and the simulation results are averaged over 10 5 independent channel realizations Unless explicitly mentioned, we assume that there are L = 5 relay nodes in the interference MIMO relay system As a benchmark, we compare the performance of the proposed algorithm with the naive AF algorithm with the source and relay precoding matrices are scaled identity matrices In the first example, we study the performance of the proposed algorithm at different number of iterations Fig 2 shows the performance of the proposed algorithm at different number of iterations for the first source-destination pair k = 1 It can be seen from Fig 2 that the proposed algorithm yields a much smaller than the naive AF algorithm, since the source and relay precoding matrices are not optimized in the naive AF algorithm It can also be observed from Fig 2 that the system reduces with increasing number of iterations During simulations, we observed that after 20 iterations, the decreasing of the objective function is negligible Thus, we suggest that only 20 iterations are needed to achieve good performance For this example, the of each source-destination pair versus P at 10 iterations is shown in Fig 3 It can be seen Copyright C 2014 by IEICE 643

5 Proposed It 5 Proposed It 10 Proposed It 20 Naive AF P db Proposed 5 relays Proposed 10 relays Naive AF 5 relays P db Fig 2: Example 1: versus P at different number of iterations Fig 4: Example 2: versus P for different L Proposed user 1 Proposed user 2 Proposed user 3 Naive AF P db Fig 3: Example 1: versus P for each source-destination pair that all three source-destination pairs achieve almost identical, indicating that the proposed algorithm is fair to all links In the second example, we study the performance of the proposed algorithm with different number of relay nodes Fig 4 shows the performance of the proposed algorithm at 10 iterations with L = 5 and L = 10 It can be seen that by increasing the number of relay nodes, the system spatial diversity is increased, and thus, a better performance is achieved In particular, we observe that a 10dB gain is obtained at the of by increasing L from 5 to 10 V CONCLUSION We have investigated transceiver design for interference MIMO relay systems based on the MMSE criterion An iterative algorithm has been developed to jointly optimize the source, relay, and receiver matrices under power constrains at each source node and relay node Numerical simulation results show that this algorithm converges quickly after a few iterations and has better performance than the existing technique ACKNOWLEDGMENT This research was supported under the Australian Research Council s Discovery Projects funding scheme DP REFERENCES [1] Y Fan and J Thompson, MIMO configurations for relay channels: Theory and practice, IEEE Trans Wireless Commun, vol 6, pp , May 2007 [2] L Sanguinetti, A A D Amico, and Y Rong, A tutorial on the optimization of amplify-and-forward MIMO relay systems, IEEE J Selet Areas Commun, vol 30, pp , Sep 2012 [3] G Kramer, M Gastpar, and P Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Trans Inf Theory, vol 51, pp , Sep 2005 [4] Y Rong, X Tang, and Y Hua, A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems, IEEE Trans Signal Process, vol 57, pp , Dec 2009 [5] C Zhao and B Champagne, Joint design of multiple non-regenerative MIMO relaying matrices with power constraints, IEEE Trans Signal Process, vol 61, pp , Oct 2013 [6] M Maddah-Ali, A Motahari, and A Khandani, Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis, IEEE Trans Inf Theory, vol 54, pp , Aug 2008 [7] B Nourani, S Motahari, and A Khandani, Relay-aided interference alignment for the quasi-static X channel, in Proc IEEE Int Symposium Inf Theory, Seoul, Korea, Jun 28-Jul 3, 2009, pp [8] X Wang, Y-P Zhang, P Zhang, and X Ren, Relay-aided interference alignment for MIMO cellular networks, in Proc IEEE Int Symposium Inf Theory, Cambridge, MA, Jul 1-6, 2012, pp [9] M Khandaker and Y Rong, Interference MIMO relay channel: Joint power control and transceiver-relay beamforming, IEEE Trans Signal Process vol 60, pp , Dec 2012 [10] D P Palomar, J M Cioffi, and M A Lagunas, Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization, IEEE Trans Signal Process, vol 51, pp , Sep 2003 [11] S M Kay, Fundamentals of Statistical Signal Processing: Estimation Theory Englewood Cilffs, NJ: Prentice Hall, 1993 [12] J W Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans Circuits Syst, vol 25, pp , Sep 1978 [13] S Boyd and L Vandenberghe, Convex Optimization Cambridge, U K: Cambridge University Press, 2004 [14] M Grant and S Boyd, Cvx: Matlab software for disciplined convex programming webpage and software, [Online] Available: Apr 2010 [15] Y Nesterov and A Nemirovski, Interior Point Polynomial Algorithms in Convex Programming Philadelphia, PA: SIAM, 1994 Copyright C 2014 by IEICE 644